dz + η 1 r r 2 + c 1 ln r + c 2 subject to the boundary conditions of noslip side walls and finite force over the fluid length u z at r = 0


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1 Poiseuille Flow Jean Louis Maie Poiseuille, a Fench physicist and physiologist, was inteested in human blood flow and aound 1840 he expeimentally deived a law fo flow though cylindical pipes. It s extemely useful fo all kinds of hydodynamics such as plumbing, flow though hypedemic needles, flow though a dinking staw, flow in a volcanic conduit, etc. Fo this eason, it is geneally known as pipe flow. Actually, the cgs unit of viscosity, the Poise P, was named afte Poiseuille and is still used in many engineeing texts. A single Poise is equivalent to 10 Pas the SI unit, thus making the Pas measue of viscosity the peso of fluid dynamics and Poise equivalent to the dolla. This is the fist of many special cases of NavieStokes equation in which vey simplified situations can be solved analytically. Pipe flow is defined to be unidiectional, i.e. thee is only a single nonzeo component of velocity and that component is both independent of distance in the flow diection and has the same diection eveywhee. The geomety is that of a long cylindical pipe with length l and adius a so the appopiate coodinate system is cylindical pola, θ, z. The pessues at each end of the pipe ae P 1 and P 0 so the pessue gadient, dp/, is constant eveywhee in the pipe. The unidiectional natue of the poblem means u and u θ, thus the continuity equation is educed to uz. This means that because z of the incompessibility constaint, at any value of z the velocity must both be a constant value as well as have an identical velocity pofile. Futhemoe, any change in the flow will occu eveywhee in the pipe instantaneously. Of couse, you aleady know this is tue because you have taken a showe without a pessue egulato so that when somebody else flushes a toilet and cold wate is diveted to efill the toilet tank, the pessue gadient in the pipes fo cold wate dops, deceasing the flow of cold wate and exposing you to the hot wate alone ouch!! Howeve, even in the moe geneal case of the NavieStokes equation that has an inetial tem, ρ u z + uz t z, one can see that fo steady flow u z the geomety of the poblem and the t incompessibility of the fluid specify that the inetial tem is exactly zeo. So Poiseuille Flow is not limited to the Stokes egime, but also occus at highe Re and we ll see that this is impotant. This 1D vesion of the momentum equation in cylindical coodinates is then We will ty a solution of the fom dp + η u z o dp + η 1 u z = 1 dp 4η + c 1 ln + c u z 1 subject to the bounday conditions of noslip side walls and finite foce ove the fluid length u z at u z at = a 3 Solving fo the constants we now have u z = 1 dp 4η a 4 This means the velocity pofile of the flow has a paabolic shape with a maximum in the cente and is zeo at the pipe walls. Also note that the flow is independent of the fluid density. 1
2 As mentioned, the velocity is maximum at the cente, which we can calculate u max = dp a 4η at 5 Pessue gadients ae nomally defined to be negative, such that wate flows fom high pessue to low pessue, so when P 1 > P 0, u max is a positive quantity. It is also useful to calculate the total flow ate though the pipe, so we integate the velocity ove the a cosssection of the pipe Q = a 0 π ud = a 0 π 1 dp 4η a d = dp πa 4 8η = πa4 P 0 P 1 8ηl 6 The volumetic flow ate units of volume/time o m 3 /s shows that fo a given pessue gadient and viscosity, the flow though the pipe is popotional to the adius of the pipe to the fouth powe. This is what Poiseuille demonstated expeimentally. The mean velocity is simply the total flow nomalized by the cosssectional aea of the pipe ū = dp πa 4 8πa η = dp a 8η = 1 u max 7 The mean velocity is the esult of the net foce exeted on the fluid by the pessue gadient acting to ovecome the viscous dag fom the pipe walls. The foce pe unit length fom pessue is F P = πa P 0 P 1 l = πa dp This shows that the mean flow, ū, is elated to the pessue foce by ū = F P /8πη and so it is linealy invesely popotional to η. Similaly, fo a Newtonian fluid, viscous dag is popotional to the shea tangential stess, σ z, which we can evaluate nea the wall of the pipe, = a σ z =a = η u z =a = η a dp η = a dp Simila to the nondimensional dag coefficient of the Stokes sphee, c D, we can detemine a fiction facto, f, which descibes the effect of dag. We use the shea stess evaluated at the wall as a chaacteistic stess and nomalize that value by a chaacteistic pessue 1ρ fū in which we use the mean velocity f = σ z =a 1 ρ fū = 4a dp 10 ρ f ū If we substitute fo just one of the ū, then f looks like f = 4a 1 dp ρ f ū ū = 4a ρ f ū 8η a dp dp = 3η ρ f ūa If we choose a chaacteistic length scale as the diamete of the pipe, D = a, then we have f = 64η ρ f ūd = 64 Re 1 This elationship holds until the tansition into the tubulent flow egime at Re
3 Channel Flow Anothe unidiectional flow is the flow between two igid plates diven by a pessue gadient. This is actually just Poiseuille flow in Catesian geomety with ẑ the same diection as in cylindical pola so the pessue gadient and esultant flow ae both only in the x diection u y = u z and the velocity pofile vaies with z. The geomety has the xaxis along the midplane of the channel, and since the channel has height h, the channel walls ae at ±h/. The govening equations ae u x x dp + dx η u x o dp + η u x dx z Since dp/dx is constant, this is a second ode O.D.E. and the integation is staightfowad 13 u x = 1 dp η dx z + c z + c 1 14 The bounday conditions ae fom the mio symmety along the midplane u x z = u x z and noslip at the walls u x z = ±h We can now solve fo the constants of integation and get the velocity pofile. All the same insights fom Poiseuille flow in a pipe ae applicable hee. u x = 1 dp [ z h/ ] 15 η dx The velocity pofile is again paabolic in shape and constant eveywhee. Couette Flow Couette flow is simila to channel flow and has the same geomety but with an impotant modification. Instead of the pessue gadient diving the flow, it is diven by the motion of one of the boundaies and that motion is paallel to the diection of the channel dp is in fact absent fom dx this poblem. The assumption is that some extenal foce is applied to move the wall and that applied foce simply scales with the viscosity of the fluid. Depending on the efeence fame you choose to do the poblem in, the top o bottom plate can be moving at some velocity U 0 o they can both move in opposite diections at U 0 /. The most convenient choice fo the coodinate system is to have a stationay plate at z and a moving plate at z = h so again the channel has height h. The govening equations fo a shea diven flow ae even simple than fo channel flow since now dp dx u x x 16 0 = η u x o 0 = η u x z Twice integating this second ode O.D.E. gives the solution u x = c 1 z + c. The bounday conditions ae again noslip velocity bounday conditions at the stationay and moving walls, so u x z and u x z = h = U 0. The solution fo the u x is again a constant z u x = U 0 17 h The velocity pofile in a shea diven flow is again identical fo all values of x, vaies linealy with distance fom the moving wall, and is independent of both density and viscosity. Also note that the shea stess is also constant eveywhee σ xz = η u x z = η U 0 h 18 3
4 Classification of PDEs and types of Bounday Conditions Any PDE can be classified using the method of chaacteistics which detemines if the PDE is eithe hypebolic, elliptic, o paabolic. Both Laplace s equation and Poisson s equation ae classified as elliptical, and is a common class of equation one encountes in fluid dynamics. Othe examples include of the wave equation hypebolic and the diffusion equation paabolic. It is impotant to undestand which class of equation you ae attempting to solve, in paticula if you ae using numeical methods, because the stability o success of the numeical method applied to one class of equation may be a completely unstable o be an unsuccessful appoach if applied to a diffeent class of PDE. The pimay vaiable is the vaiable in the govening equation eithe PDE o ODE and evey pimay vaiable always has an associated seconday vaiable. The seconday vaiable is usually the deivative of the pimay vaiable and always has a physical meaning that is often a quantity of inteest. In fluid dynamics the pimay vaiable is velocity and the seconday vaiable is stess. Anothe example is heat tansfe in which the pimay vaiable is tempeatue and the seconday vaiable is heat flux. In ode to obtain a solution to any PDE, bounday conditions must be specified. Thee ae two types of bounday conditions that can be applied: those that specify the pimay dependent vaiable on bounday and those that specify a seconday vaiable on the bounday, and usually the deivative is taken nomal to the bounday. The fist type of bounday condition is called an essential bounday condition and when solving an elliptic class of equation it is known as a Diichlet bounday condition. The second type of bounday condition is called a natual bounday condition and when solving an elliptic class of equation it is known as a Neumann bounday condition. It is quite ok, and even somewhat common, to have mixed types of bounday conditions along diffeent pats of the bounday. Fo example, one potion of the bounday will specify a Diichlet bounday condition and anothe potion will specify a Neumann bounday condition. Howeve, it is impossible to specify both types of conditions at the same point of any potion of the bounday. Thus, if the tempeatue is specified, the heat flux will be detemined o vicevesa but it can neve happen that both ae specified at the same place. Similaly, if the stess is specified at a given point, the velocity will be solved fo on the bounday at the same point. This is actually quite a poweful, and useful, thing to know, especially in situations like Couette flow and channel flow, which have the same geomety. It is actually possible to combine the simple solutions fom both poblems because 1 they ae both linea ODEs we can use the pinciple of supeposition and the solutions wee aived upon by applying the same type of bounday condition. Both poblems specified the velocity on the walls and theefoe both applied Diichlet bounday conditions. We can then wite the solution of Couette flow that now includes a pessue gadient by simply tansfoming the channel flow solution to a coodinate system with the bottom wall at z so z u x = U 0 h + 1 dp [ z hz ] 19 η dx A simple model of asthenospheic counteflow is motivated by a shea flow diven by plate motions on the suface. The shea flow sets up a pessue gadient in the the opposite diection which dives an associated channel flow undeneath the shea flow a etun flow. This is the same as the above poblem, except the diection of the pessue gadient is evesed z u x = U 0 h 1 dp [ z hz ] 0 η dx 4
5 Steam Function The steam function, ψ, is both an illustative and useful appoach to apply to fluid dynamics as it can povide elatively quick solutions to D incompessible flow poblems. The majo dawback of the steam function is that it is basically limited entiely to D incompessible flow poblems. The steam function is like a potential field in that only the diffeence in ψ between two points has any physical meaning the absolute value of ψ is abitay. Lines of constant ψ ae called steam lines and give an excellent visual epesentation of the flow, howeve, only in a D geomety. In D, the incompessibility constaint is fom the continuity equation The definition of the steam function is then u o u x x + u y y 1 u x = ψ y u y = ψ x The steam function satisfies the continuity equation ψ x y + ψ y x 3 The steam function can also be substituted into the Stokes equation 0 = dp η 3 ψ + 3 ψ dx x y 3 y 0 = dp dy + η 4 3 ψ + 3 ψ 3 x y x Now eliminate the pessue tem using the same technique that was applied ealie when solving fo the flow aound a Stokes sphee, i.e. take patial deivatives w..t. the othe dimension [ ] 0 = dp η 3 ψ + 3 ψ y dx x y 3 y [ ] 5 0 = dp + η 3 ψ + 3 ψ x dy 3 x y x and then subtacting the esulting equations we get 0 = 4 ψ 4 x + 4 ψ x y + 4 ψ 4 y Reaanging the deivatives we now have 0 = x + y x + y 6 ψ 7 This equation can be ecognized as the Laplacian opeato being applied twice to ψ 0 = ψ 8 And this is known as the Bihamonic opeato = 4 which we can use to wite 0 = 4 ψ 9 Thee ae wellknown solutions to this equation and it is also valid fo noncatesian geometies. 5
6 Cone Flow The situation of a subduction zone is in some ways analogous to one vaiation of the classic cone flow poblem in fluid dynamics. In this vesion, two igid plates infinite in extent convege at a point whee the advancing plate plate A dips at an angle below the backac plate plate B. We will use the point of convegence as the oigin of a D cylindical coodinate system with plates on the suface the line at θ. The angle that plate A makes between itself on the suface and the dipping potion is defined as θ a and the dip angle between plates A and B is defined as θ b and assumed to be acute. Plate B is assumed to emain stationay while plate A is moving on the suface at velocity u = U 0 towads the oigin and along the dip angle at u = U 0 away fom the oigin. Fo both plates, the velocities in the θ diection ae assumed to be zeo u θ. Notice that thee ae no body foces in this poblem, and that the Stokes flow is diven entiely by the velocity bounday conditions which themselves ae diving by some applied foce but since it is not a body foce it is ielevant. The govening equations fo Stokes flow ae simply τ and u. Expanding the momentum equation out into the components of total stess τ τ θ τ θ θ τ θθ θ We can use the constitutive elationship between total stess and stain ate, τ = P I + ηd 30 τ = P + σ = P + η ε τ θθ = P + σ θθ = P + η ε θθ 31 τ θ = τ θ = σ θ = η ε θ And ewite the total stess with the stain ate having tems of the velocity gadients τ = P + η u τ θθ = P + η 1 u θ θ τ θ = η 1 u θ + u θ + u u θ Notice that if we add the nomal components of stess togethe we get u τ + τ θθ = P + η + 1 u θ θ + u The nd tem on the RHS vanishes since u, and because the fluid is isotopic the expessions fo nomal stesses become τ = P τ θθ = P Using these allows us to expess the momentum equation entiely in tems of P and τ θ P + 1 τ θ θ 1 P + τ θ θ 35 6
7 and this can be ewitten as P + η 1 1 P θ + η 1 u θ θ 1 u θ + u θ + u θ u θ u θ 36 Seems all that manipulation didn t help as the momentum equation still looks a little ugly. Luckily, it was shown that solving 4 ψ will also give the solution fo velocity, and it tuns out the steam function is a moe convenient way to appoach the poblem. In D cylindical, the Laplacian is ψ = 1 ψ + 1 ψ 37 θ Consideing the geomety of the poblem has plates of infinite extent with constant elative velocity, the solution fo velocity eveywhee is expected to be independent of. This means the equation is sepaable and we will use a solution of the fom Simple substitution fo ψ gives ψ = RT θ and 38 u = 1 ψ θ u θ = ψ u = 1 θ R T 40 θ which means R = and then ψ = T θ which upon substituting back into the Bihamonic equation gives 4 T θ + T 4 θ + T 41 The 4 th ode PDE has now been educed to a 4 th ode ODE which has a geneal solution of the fom T θ = A sin θ + B cos θ + Cθ sin θ + Dθ cos θ 4 and thee ae also 4 bounday conditions but these ae given as velocities so we need u and u θ 39 u = T θ θ = A cos θ B sin θ + Csin θ + θ sin θ + Dcos θ θ sin θ u θ = T θ 43 At this point its a good idea to beak the poblem into two potions and solve fo the steam function in each domain. The obvious choice fo the two domains is the backac egion fomed by the acute dip angle between the subducting plate and oveiding plate and the foeac egion undeneath the subducting plate. The flows ae identical along the bounday of the subducting plate, and this line is known as the sepaatix. The bounday conditions ae then u θ = U 0 in the foeac egion u θ in the backac egion u θ θ in both egions u θ θ = θ b along the sepaatix u θ θ = θ b = U 0 along the sepaatix 44 7
8 Each egion has 4 bounday conditions to solve fo the 4 unknowns constants, and afte a lot of algeba one aives at the solution ψ a = U 0[θ a θ sin θ θ sinθ a θ] θ a+sin θ a ψ b = U 0[θ b θ sin θ b sin θ θ b θ sinθ b θ] θ b sin θ b o moe simply, ψ a = U 0 f a θ in the foeac egion o moe simply, ψ b = U 0 f b θ in the backac egion The velocities in each egion ae eadily obtained though diffeentiation of ψ: u = U 0 f aθ and u θ = U 0 f a θ in the foe ac and u = U 0 f b θ and u θ = U 0 f b θ in the back ac. In ode to obtain the pessue, we need to go back to the momentum equation and use the fact that τ = τ θθ = P which itself is elated to the shea tangential stess τ θ = η U 0 [ f a θ f a θ] τ θ = η U 0 [f a θ + f a θ] This helps obtain the pessue solution P a, θ = U 0η P b, θ = U 0η [sin θ sinθ θ a] θ a+sin θ a [θ b sinθ b θ sin θ b sin θ] θ b sin θ b in the foeac egion in the backac egion in the foeac egion in the backac egion Inspection of these solutions eveals that P a is always a positive quantity and P b is always a negative quantity. A positive pessue below the subducting plate implies compession o upwad foce on the suface. A negative pessue in the mantle wedge indicates that thee is a suction between the subducting plate and oveiding plate. This cone flow suction acts as a hydodynamic lift that is popotional to the pessue diffeence above and below the slab. The lift is found by integating P, θ along the dip angle, θ a, ove a length l. The toque exeted by lift is balanced by gavity though the weight of the slab with thickness h and density ρ. T flow = [ ] l [P sin θ 0 a, θ a P b, θ b ] d = U 0 ηl b π θ b +sin θ b + sin θ b θb sin θ b 48 T gavity = 1 ρghl cos θ b Both toques can be nomalized by a chaacteistic toque, U 0 ηl, which then allows one to find the citical dip angle, θ c that detemines when the toque deived fom gavity is balanced by the lift geneated by ciculation in the mantle wedge. Fo any angle smalle than θ c, the toque exeted on the slab by mantle flow will exceed the weight of the slab, and assuming the velocities emain constant, a positive feedback will occu such that θ deceases to zeo. This citical angle was detemined by Stevenson and Tune, 1977, to be 63 fo which they found the net toque was about times the chaacteistic toque. Assuming a 100 km thick slab that is 600 km in length subducts at 6 cm/y and has ρ =80 kg/m 3 gives ρghl 4ηU 0 which can be used to estimate the uppe mantle viscosity, η = π 10 1 Pa s 49 Clealy, θ c = 63 is too lage as many slabs ae obseved to have dip angles shallowe than this estimate, so obviously thee must be many othe impotant factos. One of the moe impotant factos is the nonnewtonian heology of the mantle wedge as studied by Tovish et al who found this educed θ c = 54 fo a powe law fluid with n=3. Thee ae also easons fo θ c to be lage, as slabs with finite lateal extent allow fo a 3D component of the mantle flow i.e the tooidal flow aound slab edges which educes the pessue diffeential Dvokin et al
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